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## Archive of Issues

Russia Yekaterinburg
Year
2017
Volume
27
Issue
2
Pages
210-221
 Section Mathematics Title On approximate solvability set construction in a guidance problem for a time-invariant control system on a finite time interval Author(-s) Parshikov G.V.a Affiliations Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa Abstract A time-invariant control system on a finite time interval in the finite-dimensional Euclidean space is considered. We discuss a problem of guidance with a compact target set for a control system on a given time interval. One way to solve the considered guidance problem is based on finding a solvability set in the phase space, namely, a set of all system positions from which, as from the initial ones, the guidance problem is solvable. The construction of the solvability set is an independent time-consuming problem which rarely has an exact solution. In this paper we discuss the approximate construction of a solvability set in the guidance problem for a time-invariant nonlinear control system. It is well-known that this problem is closely connected with the problem of constructing integral funnels and trajectory tubes of control systems. Integral funnels of control systems can be approximately constructed step-by-step as sets of corresponding attainability sets, therefore, attainability sets are considered to be the basic elements of the solving construction in this paper. Here, we propose a scheme of the solvability set approximate construction in a guidance problem for a time-invariant control system on a finite time interval. The basis of this scheme is reduction to the solvability sets approximate calculation of a finite number of simpler problems, namely, problems of guidance with the target set at fixed time moments from the given time interval. Wherein, the moments of time have to be chosen quite tightly in the mentioned time interval. As an example, we provide mathematical modeling of the guidance problem of the control system named “Translational Oscillator with Rotating Actuator” as well as the graphical support of the problem solution. Keywords control system, guidance problem, attainability set, solvability set, solvability set approximation UDC 517.977.58 MSC 37M05, 49M25, 93C15 DOI 10.20537/vm170205 Received 16 May 2017 Language Russian Citation Parshikov G.V. On approximate solvability set construction in a guidance problem for a time-invariant control system on a finite time interval, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 2, pp. 210-221. References Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial’nye igry (Positional differential games), Moscow: Nauka, 1974, 456 p. Kurzhanski A.B. Upravlenie i nablyudenie v usloviyakh neopredelennosti (Control and observation under uncertainty), Moscow: Nauka, 1977, 392 p. Krasovskii N.N. Upravlenie dinamicheskoi sistemoi (Control of dynamic system), Мoscow: Nauka, 1985, 520 p. Kurzhanski A.B. Izbrannye trudy (Selected papers), Moscow: Moscow State University, 2009, 756 p. Ushakov V.N., Matviichuk A.R., Parshikov G.V. A method for constructing a resolving control in an approach problem based on attraction to the feasibility set, Proceedings of the Steklov Institute of Mathematics, 2014, vol. 284, suppl. 1, pp. 135-144. DOI: 10.1134/S0081543814020126 Chernousko F.L. Otsenivanie fazovogo sostoyaniya dinamicheskikh sistem: Metod ellipsoidov (Dynamical systems phase state estimation: ellipsoid method), Moscow: Nauka, 1988, 319 p. Kurzhanskii A.B., Valyi I. Ellipsoidal calculus for estimation and control, Boston: Birkhäuser, 1997, 321 p. Lempio F., Veliov V. Discrete approximation of differential inclusions, Bayreuther Mathematische Schriften, 1998, vol. 54, pp. 149-232. Guseinov Kh.G., Moiseev A.N. Ushakov V.N. The approximation of reachable domains of control systems, Journal of Applied Mathematics and Mechanics, 1998, vol. 62, issue 2, pp. 169-175. DOI: 10.1016/S0021-8928(98)00022-7 Gusev M.I. Estimates of reachable sets of multidimensional control systems with nonlinear interconnections, Proceedings of the Steklov Institute of Mathematics, 2010, vol. 269, suppl. 1, pp. 134-146. DOI: 10.1134/S008154381006012X Filippova T.F. Construction of set-valued estimates of reachable sets for some nonlinear dynamical systems with impulsive control, Proceedings of the Steklov Institute of Mathematics, 2010, vol. 269, suppl. 1, pp. 95-102. DOI: 10.1134/S008154381006009X Matviychuk A.R., Ukhobotov V.I., Ushakov A.V., Ushakov V.N. A guidance problem for a nonlinear control system on a finite time interval, Prikl. Mat. Mekh., 2017, vol. 81, issue 2, pp. 165-187 (in Russian). Khalil H.K. Nelineinye sistemy (Nonlinear systems), Izhevsk: Regular and Chaotic Dynamics, 2009, 812 p. Full text